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Generating sequences

Normal sequence

x <- seq(1:10)
x
 [1]  1  2  3  4  5  6  7  8  9 10

Odd numbers only

x.odd <- seq(1,10,2)
x.odd
[1] 1 3 5 7 9

Even distribution between ranges. Example: 3 equal division between 1 and 10.

y.divide <- seq(1,10, length.out = 3)
y.divide
[1]  1.0  5.5 10.0

Monte Carlo Simulations

Discrete

Monte Carlo simulations model the probability of different outcomes by repeating a random process a large enough number of times that the results are similar to what would be observed if the process were repeated forever. First, create an box with data. This can be done through rep (2 red, 3 blue balls:

beads <- rep(c("red", "blue"), times = c(2,3))
beads
[1] "red"  "red"  "blue" "blue" "blue"

Lets pick a bead from this box 10,000 times. To do this, we set up B and run it.

B <- 100000    
events <- replicate(B, sample(beads, 1))    # draw 1 bead, B times
tab <- table(events)    # make a table of outcome counts
prop.table(tab)    # view table of outcome proportio
events
   blue     red 
0.60104 0.39896

Continuous

Generating simulated height data using normal distribution.

x <- heights %>% filter(sex=="Male") %>% pull(height)
n <- length(x)
avg <- mean(x)
s <- sd(x)
simulated_heights <- rnorm(n, avg, s)

The sumulated_heights is built from the length, mena, and sd. The plot of this is as follows. Important to note that this is built from the assumption of normal distribution:

data.frame(simulated_heights = simulated_heights) %>%
    ggplot(aes(simulated_heights)) +
    geom_histogram(color="black", binwidth = 2)

Example, running a monte carlo simulation to find out the probablity of people over 7 feet (answer: 0.0214). Running the simulation 10,000 times (B):

set.seed(3233)
B <- 10000
tallest <- replicate(B, {
    simulated_data <- rnorm(800, avg, s)    # generate 800 normally distributed random heights
    max(simulated_data)    # determine the tallest height
})
mean(tallest >= 7*12) #7*12 = 84 inches, which is 7 feet. 
[1] 0.0214

Important to note the set.seed() function allows us to control the random. Without this function, the answer would be different because you cannot control the random variability. But with set.seed() you will have the same answer every time.

Mean / Sd

For mean, median, sd, min, max:

heights %>%
    filter(sex == "Male") %>%
    summarize(average=mean(height), sd=sd(height), median = median(height),
              minimum = min(height), maximum = max(height))

Observing Columns

murders is the dataset. To observe the column types,

str(murders)
'data.frame':   51 obs. of  5 variables:
 $ state     : chr  "Alabama" "Alaska" "Arizona" "Arkansas" ...
 $ abb       : chr  "AL" "AK" "AZ" "AR" ...
 $ region    : Factor w/ 4 levels "Northeast","South",..: 2 4 4 2 4 4 1 2 2 2 ...
 $ population: num  4779736 710231 6392017 2915918 37253956 ...
 $ total     : num  135 19 232 93 1257 ...

To observe all column numbers. Easy to see which column is which number.

names(murders)
[1] "state"      "abb"        "region"     "population" "total"

To observe the first 5 data data of a column.

murders$state[1:5]
[1] "Alabama"    "Alaska"     "Arizona"    "Arkansas"   "California"

Data Frames (Vectors)

Associating vectors only

city <- c("Tokyo", "Lile", "Dover")
area <- c(10, 8, 13)
names(area) <- city
area
Tokyo  Lile Dover 
   10     8    13

Creating Data Frames. Extra column created because of previous code.

data <- data.frame(name=city, value=area)
data
       name value
Tokyo Tokyo    10
Lile   Lile     8
Dover Dover    13

Normal Distribution

q-q graph

To see if male dataset follow a normal distribution. First define p as quantile ranges from 0.05 to 0.95.

p <- seq(0.05, 0.95, 0.05)

The observed_q is the real quantile of your dataset. The theory_q is the expected quantiles of a normal distribution.

observed_q <- quantile(male$height, p)
theory_q <- qnorm(p, mean=mean(male$height), sd=sd(male$height))

The see closely they match, simply plot them. This shows that points almost match, so male dataset is a good approximation for normal distribution. ***


sessionInfo()
R version 4.0.0 (2020-04-24)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Catalina 10.15.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] dslabs_0.7.3    forcats_0.5.0   stringr_1.4.0   dplyr_0.8.5    
 [5] purrr_0.3.4     readr_1.3.1     tidyr_1.0.3     tibble_3.0.1   
 [9] ggplot2_3.3.0   tidyverse_1.3.0 workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] tidyselect_1.1.0 xfun_0.13        haven_2.2.0      lattice_0.20-41 
 [5] colorspace_1.4-1 vctrs_0.3.0      generics_0.0.2   htmltools_0.4.0 
 [9] yaml_2.2.1       rlang_0.4.6      later_1.0.0      pillar_1.4.4    
[13] withr_2.2.0      glue_1.4.1       DBI_1.1.0        dbplyr_1.4.3    
[17] modelr_0.1.7     readxl_1.3.1     lifecycle_0.2.0  munsell_0.5.0   
[21] gtable_0.3.0     cellranger_1.1.0 rvest_0.3.5      evaluate_0.14   
[25] labeling_0.3     knitr_1.28       httpuv_1.5.2     fansi_0.4.1     
[29] broom_0.5.6      Rcpp_1.0.4.6     promises_1.1.0   backports_1.1.6 
[33] scales_1.1.1     jsonlite_1.6.1   farver_2.0.3     fs_1.4.1        
[37] hms_0.5.3        digest_0.6.25    stringi_1.4.6    grid_4.0.0      
[41] rprojroot_1.3-2  cli_2.0.2        tools_4.0.0      magrittr_1.5    
[45] crayon_1.3.4     whisker_0.4      pkgconfig_2.0.3  ellipsis_0.3.0  
[49] xml2_1.3.2       reprex_0.3.0     lubridate_1.7.8  rstudioapi_0.11 
[53] assertthat_0.2.1 rmarkdown_2.1    httr_1.4.1       R6_2.4.1        
[57] nlme_3.1-147     git2r_0.27.1     compiler_4.0.0